Do variational formulations for inhomogeneous density functions lead to unique solutions?

Kurtosis is a commonly used descriptive statistics. Kurtosis “Coefficient of excess” is critically reviewed in different aspects and is called as, measuring the fatness of the tails of the density functions, concentration towards the central value, scattering away from the target point or degree of peakedness of probability distribution. Kurtosis is referred to the shape of the distribution but many distributions having same kurtosis value may have different shapes while Kurtosis may exist when peak of a distribution is not in existence. Through extensive study of kurtosis on several distributions, Wu (2002) introduced a new measure called “W-Peakedness” that offers a fine capture of distribution shape to provide an intuitive measure of peakedness of the distribution which is inversely proportional to the standard deviation of the distribution. In this paper the work is extended for different others continuous probability distributions. Empirical results through simulation illustrate the proposed method to evaluate kurtosis by W-peakedness

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Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

Remote Sensing ◽

10.3390/rs13122307 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2307

Author(s):

J. Javier Gorgoso-Varela ◽

Rafael Alonso Ponce ◽

Francisco Rodríguez-Puerta

Keyword(s):

Probability Density ◽

Linear Models ◽

Pinus Halepensis ◽

Beta Function ◽

Probability Density Functions ◽

Lidar Data ◽

Density Functions ◽

Minimum Diameter ◽

Location Parameters ◽

Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

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A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab087 ◽

2021 ◽

Vol 502 (2) ◽

pp. 1768-1784

Author(s):

Yue Hu ◽

A Lazarian

Keyword(s):

Magnetic Fields ◽

Probability Density ◽

Column Density ◽

Mass Density ◽

Probability Density Functions ◽

Emission Lines ◽

Density Functions ◽

Star Forming ◽

Velocity Gradients ◽

Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

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A numerical study of gravity-driven instability in strongly coupled dusty plasma. Part 1. Rayleigh–Taylor instability and buoyancy-driven instability

Journal of Plasma Physics ◽

10.1017/s0022377821000349 ◽

2021 ◽

Vol 87 (2) ◽

Author(s):

Vikram S. Dharodi ◽

Amita Das

Keyword(s):

Dusty Plasma ◽

Numerical Study ◽

Fluid Model ◽

Density Region ◽

Strongly Coupled ◽

Rising Bubble ◽

Inhomogeneous Density ◽

Gravity Driven ◽

The Individual ◽

Strongly Coupled Dusty Plasma

Rayleigh–Taylor (RT) and buoyancy-driven (BD) instabilities are driven by gravity in a fluid system with inhomogeneous density. The paper investigates these instabilities for a strongly coupled dusty plasma medium. This medium has been represented here in the framework of the generalized hydrodynamics (GHD) fluid model which treats it as a viscoelastic medium. The incompressible limit of the GHD model is considered here. The RT instability is explored both for gradual and sharp density gradients stratified against gravity. The BD instability is discussed by studying the evolution of a rising bubble (a localized low-density region) and a falling droplet (a localized high-density region) in the presence of gravity. Since both the rising bubble and falling droplet have symmetry in spatial distribution, we observe that a falling droplet process is equivalent to a rising bubble. We also find that both the gravity-driven instabilities get suppressed with increasing coupling strength of the medium. These observations have been illustrated analytically as well as by carrying out two-dimensional nonlinear simulations. Part 2 of this paper is planned to extend the present study of the individual evolution of a bubble and a droplet to their combined evolution in order to understand the interaction between them.

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Variational formulations of steady rotational equatorial waves

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0146 ◽

2020 ◽

Vol 10 (1) ◽

pp. 534-547

Author(s):

Jifeng Chu ◽

Joachim Escher

Keyword(s):

Linear Stability ◽

Stream Function ◽

Water Waves ◽

Variational Formulation ◽

Energy Functional ◽

Equatorial Waves ◽

Second Variation ◽

Governing Equations ◽

The Second Variation ◽

Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

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Modelling with star-shaped distributions

Dependence Modeling ◽

10.1515/demo-2020-0003 ◽

2020 ◽

Vol 8 (1) ◽

pp. 45-69

Author(s):

Eckhard Liebscher ◽

Wolf-Dieter Richter

Keyword(s):

Probabilistic Models ◽

Arbitrary Dimension ◽

Probability Density Functions ◽

Density Functions ◽

Wide Range ◽

Gaussian Density ◽

Spherical Distributions ◽

Data Points ◽

Non Gaussian ◽

General Method

AbstractWe prove and describe in great detail a general method for constructing a wide range of multivariate probability density functions. We introduce probabilistic models for a large variety of clouds of multivariate data points. In the present paper, the focus is on star-shaped distributions of an arbitrary dimension, where in case of spherical distributions dependence is modeled by a non-Gaussian density generating function.

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